Every Unsolved Geometry Problem that Sounds Easy

[@ThoughtThrill365]

Let me know if there's a topic you'd like me to cover next. 😊

[@Poggersvale]

3x+1

[@Poggersvale]

Maybe

[@MatthewJohnson-hi2th]

You should talk about your plagiarism next!

[@dlju8355]

I’m lost in the woods and I’m very glad I found this survival video.

[@KeenRunner]

Stay put! 😮

[@chri-k]

i hope it wasn't an equilateral triangle

[@TheBcoolGuy]

@@KeenRunnerHonestly, that isn't always the right answer. No point staying and starving to death and if the forest isn't that big or you know in which direction it's located relative to the compass, you can use the sun (rises in east, sets in west) to figure out where to go.

[@a308sniper]

Generally speaking there's very few places you want to actually stand still. At least in the US. Generally speaking unless you're in a very remote part of the us you want to walk in one consistent direction then hit a road. You need to know how to travel in the same direction but generally unless you're in areas where this doesn't work you're not likely to be lost. And if you're lost you just needed to not travel in a bad direction or go hike in a desert after being black out drunk. Usually within 3 days is 90% of us that people are actually in. And you can find water in most of these areas. Once you hit a road you can pick left or right and generally speaking be very near to people. Almost always you'll hit a house that you can ask for help or a driver you can flag down. But this assumes you've got somewhat knowledgeable information about where you're lost. So you'd want to do this hiking... Or, IDK you've looked at a map somewhat recently. It's not for for "stuck on an island" or ", lost in northern Canada" lvls of stuck.

[@samuelspace101]

@@TheBcoolGuy I have worked for search and rescue for my county’s sheriffs office in Oregon, always stay put, as long as at least someone knows where you are, and when you left we can find you, but in general keep a charged electronic device on you, if you can, that way if something does happen we can find you faster using your S.O.S feature or using calling your emergency service line. If you are lost or injured, don’t try to find your own way out, this is one of the fastest ways to get even more injured or even more lost, you have a much higher likelihood of getting recovered if you stay put, hopefully near a trail, if you know where you are and are not injured, that is a different situation. Currently SAR runs in all 50 states and most territories, with the exception of parts of alaska, we have the ability to make it to any location within the United Sates within 48 hours. Next time you go hiking: 1. Tell someone where you’re going, how long you will be gone, and what trails you will be taking. 2. Bring an electronic emergency device. 3. Unless you are a professional stay close to the given trail you were expecting to take. In any case you will be fine if you follow these steps.

[@autumnson]

No intro, no outro. This guy doesn't play around with our time! We love to see it!

[@weibrot6683]

How do you not know about sponsor block, intros, outros and sponsor segments haven't been an issue for years....

[@autumnson]

@@weibrot6683 My brother in christ, I already use sponsorblock. That doesn't mean you can't salute creators who actually don't make it necessary to enjoy their videos.

[@MatthewJohnson-hi2th]

It's all just taken straight from the corresponding Wikipedia articles. He doesn't cite them but if you look at the articles you'll see that a lot of the wording is just taken straight from there, but changed slightly. He's done far less work than it seems, and he doesn't credit the original authors at all. Straight up plagarism.

[@Bobo-ox7fj]

​@@MatthewJohnson-hi2th oh nyo, lucky wikipedos deserve naught but disdain

[@MatthewJohnson-hi2th]

@@Bobo-ox7fj What do you have against Wikipedia? And besides, it's still true that the creator of these videos is profiting off of the work of others. It doesn't matter if you hate the people he's stolen from, that's still dishonest and wrong, right? If an academic paper plagiarised Wikipedia would you have a problem with that?

[@bendoubleu_]

If mathmaticians dont solve them in the next year Ill have to get involved 🐺

[@Gordy-io8sb]

As if you could prove any of them.

[@Gordy-io8sb]

Also, mathematical conjectures can take eternities for proofs to be published. Just look at how long it took for someone to solve the FLT.

[@DoReMeDesign]

the chosen one 🙏

[@themathhatter5290]

​@@Gordy-io8sb Anyone can do math, it's only a matter of what tools they have available to them

[@projectasteria]

@@Gordy-io8sb clearly you haven't met ben

[@russell_rozenbaum]

GUYS. Watch every section all the way though. Ended up in a forest for several days and did not find a solution.

[@mdrdprtcl]

RIP

[@Valentin-g4i5p]

3:15 sorry😢

[@debblez]

2:51 Correction: the equilateral triangle is the only regular polygon that isn’t fat. its solution is known, but it isn’t a straight line path.

[@drilledbean7434]

well what is the solution

[@deathrobloxian]

​@@drilledbean7434zigzag line composed of 3 straight line segments of equal length

[@IamGumbyy]

If anyone else was wondering, the solution for equilateral triangles is a zig-zagging line with three segments of equal length.

[@deathrobloxian]

huh apparently can get notified despite my reply here having been deleted by something

[@Cyberpanda-ug1bf]

Nerd

[@BinaryCommando]

We are even further from solving the "Worms lost in a forest" problem

[@Roset595]

"What is the maximum packing density of worms of length = 1 in a bounded forest of arbitrary length and width, if worms cannot overlap each other or the edge of the forest?" - the next unsolved problem

[@pencilwren]

@@Roset595how wide are the worms

[@SuPeRHeRoDuDe3124]

​@@Roset595European or African worms?

[@lunondisposable5382]

@@Roset595 Sounds like the setup to an XKCD comic

[@saschabaer3327]

Having looked through the paper cited in the wikipedia article on Bellman’s lost in a forest problem, it appears that the case of the triangle is actually solved. Like, wikipedia says the equilateral triangle case is still open, but the paper they cite presents a solution. (Edit: The wikipedia article has been fixed)

[@GarryDumblowski]

Huh, that's odd. I was actually just about to ask in the comments why moving in a straight line isn't the optimal solution for a triangle.

[@saschabaer3327]

@@GarryDumblowski Simply because a better solution is known (and was proven to be optimal). It’s a zig-zag with three line segments at very specific lengths and angles that ends up being a bit shorter than a side length of the triangle, and hence better than a straight line.

[@TheBcoolGuy]

@@saschabaer3327but not worth the effort in practice. Academia moment.

[@alexismiller2349]

Obviously not meant for people in a forest. Maybe it can be used in certain search/optimisation algorithms

[@minerscale]

This is a math video lol, we don't do not worth it in practice here!

[@AA-100]

Moving Sofa problem is also an unsolved geometry problem

[@sirryanvandunn]

Well, only if you don't know how to PIVOT!!!!!!!!

[@LunarMoonPhase]

I think the moving sofa problem was solved already

[@Deathranger999]

@@LunarMoonPhaseWe have a solution that mathematicians *think* is optimal, but proving that it is is a considerably harder task.

[@jimi02468]

@@LunarMoonPhase Solving the problem means to prove that certain solution is right and I don't think that has been done. What has been done is finding a solution that mathematicians think is right. That doesn't count as a mathematical proof.

[@samuelwaller4924]

well that's just because we haven't invented time travel yet

[@minxythemerciless]

In orienteering, lost in a forest is a common thing. That is why you are given a safety bearing which is a direction you are to walk to reach a known feature such as a road where rescuers will eventually find you. The only variation is when you are too injured to walk and you sit still and blow your whistle

[@geekjokes8458]

what if the forest is triangular? /s

[@EdenCoats]

5:42 japanese flag jumpscare

[@anthonysansalone3062]

7:01 😮

[@hundvd_7]

3:51 😱

[@bodakin]

Lmfaooo I was about to comment this but yt has comments with time stamps where you are in the video

[@spiral32]

My toxic trait is believing each question will take me 5 minutes to solve

[@notenoughmice]

please do. we would all be grateful

[@dedusmuln]

i believe in you

[@20IA]

Take 5 years, for real

[@AussieGuy-qb8mh]

Same

[@eugenesis8188]

Lmfao. I was watching this, and i got really confused because i kept rewinding to figure out where the first one started. Then i realized that my brain literally stops listening when you said "square packing" because i immediately thought it was a square space ad.

[@jamesknapp64]

The fact we dont know square packing for either 11 or 17 is crazy to me. I was gonna guess 111 when some small number; and watch someone state n = 111 is actually known in the comments

[@ere4t4t4rrrrr4]

when you show something interesting, like 1:37 or some equation, i think you should show it for more than a couple seconds.. had to get back and pause the video

[@whendowedie7835]

Why, I mean I'm high but I understood it good enough

[@prydaz621]

So you're telling me I gotta move 11 box tomorrow and mathematicians can't even tell me how to arrange them ?! What are we paying them for ?

[@moth.monster]

My favorite 3D print I've made is a version of the current best 17 square packing as a puzzle. It is deeply unsatisfying and I love it for that.

[@phoule76]

I like tiling and packing problems, yet I like neither tiling nor packing.

[@potors]

3:18 instructions unclear: lost in a forest for 3 years and nobody came yet

[@alansmithee419]

7:28 This section confused me because you said the upper bound was "improved" (implying "reduced") to a bigger number. It turns out you had the initial number correct: 0.845 The other two however, you have erred by replacing the 4 with a second 8. "0.88414" should be 0.84414, and "0.88409" should be 0.84409. We also have a lower bound on this problem of 0.832, so we have quite a good answer even if it's not perfect/proven.

[@petersmythe6462]

It's kind of interesting that getting to the edge of an equalateral triangle fastest from a random point does not encourage going in a straight line.

[@Roset595]

If you don't know your position or orientation, heading in a straight line towards a corner takes longer than a straight line towards the middle of a side

[@HashPram]

I have a solution to these problems but the proof is too large to fit in the comments

[@LeynSolista]

Write a paper about It you might receive money on it

[@ravciozo2137]

Slow down there, mister Pierre de Fermat

[@jamesburke4505]

Super underrated comment

[@Roset595]

What is the minimum size of comment needed to fit your proof?

[@keremaksoy9731]

pop off hashpram

[@Zaspar]

Could you make a video about problems which went unsolved for a very long time, similar to these, but then were solved?

[@VictorSilva-lj4wy]

5:06 I thought he was starting a limerick there

[@TheIllusionaryGentleman]

Love how the video just ends, as if it was released mid-production and we can expect an update when the scholars get around to finding more impossible conundrums

[@puzzzl]

The last one is a blast, you can play with it pretty easy on a piece of paper: draw 10 lines and see if you can hit 26 triangles, the supposed limit. It's really hard to even hit 20, let alone the 25 of the best known figure (at exactly 11:26).

[@Bahelord]

6:06 the line is not on the corner

[@adamtichy8008]

THE LINE IS NOT ON THE CORNER

[@ngwoo]

Someone a couple thousand years ago accidentally dropped ten wooden dowels down a flight of stairs and picked them up without realizing they accidentally had a really interesting thing

[@SoftDevAB]

I found a truly marvelous solution to all of these problems, which this comment is to narrow to contain

[@TheScienceGuy10]

5:17 optimal packing density for balls

[@ferenc_l]

4:55 his name is misspelled - it should be Stanisław

[@ravciozo2137]

Yeah, it is so extraordinarely misspelled that I can't imagine how could someone even come up with the writing in the video

[@ThornShadowWolf]

They didn't say it was an _unbelievable_ misspelling

[@vikvavs55]

It’s like they saw a transliteration of the Russian name or saw the Czech name „Stanislav” but then did some weirdness with the caught-cot merger so they spelt it with that o but then…didn’t bother to check? Idk 😂

[@ericfox7021]

What am I misunderstanding about Lesbegue’s problem? A circle with diameter 1 is a valid cover for all shapes with diameter 1, and for a cover to be a valid cover for all shapes of diameter 1, it has to be able to cover a circle with diameter 1, therefore the smallest possible cover is a circle with diameter 1…

[@ericfox7021]

Okay, I think I found the flaw in my thinking. An equilateral triangle with diameter 1, for example, does not fit in the circle with diameter 1.

[@maxhedges4167]

Thank god I found this comment 😅 I've been going back and forth with chatgpt about this and I felt like I was going crazy lol.

[@binarycat1237]

i like the usage of proper math terms along with giving a brief definition

[@alexanderwelshwelsh9931]

The content of the exam the teacher said wasnt going to be too hard:

[@adrianclopan7892]

This video just starts. That's it: straight to the point; no chit chat

[@annaclarafenyo8185]

Kepler's conjecture is proved.

[@_arie_s]

there is a difference between a formal proof and an algorithmic justification.

[@avonbarksdale5821]

Common misconception. In fact, a conjecture can never be proven.

[@alexismiller2349]

Kepler's conjecture was proven in 1998 by Thomas Hales (well we can't be too sure since the proof was too complicated), getting officially published in 2005. Nobody was really sure about whether or not it contained an error, so in 2017, a team directed by Thomas Hales managed (with an insane amount of work) to get an entirely computer verified proof. So it has been proven, no doubt about it

[@ahdog8]

​@@avonbarksdale5821 What makes you say this? I can find no definition that says this. The Wikipedia page "List of mathematical conjectures" has a section called "conjectures now proven", leading me to believe they can be proven. Are you sure you aren't thinking of axoims?

[@tfae]

@@_arie_s 1976 called, they want their computer proof controversy back

[@TacoDude314]

Wow that last one is tantalizing

[@landsgevaer]

Lebesgue Reuleaux

[@Kettwiesel25]

He also missed an 8 after the comma for the original upper bound on the Lebesgue problem

[@Slothptimal]

For the equilateral triangle, it's walk the amount of the shortest distance from the center to the edge, then go 45° (either left or right, doesn't matter) in a straight line until out.

[@alveolate]

feels like each one can have a 6-15min video of their own too

[@samuelkress4647]

11:25 “The upper bound is 26 tringle”

[@onan-iz8cu]

I know nothing about geometry or even math tbh but I watched this with full interest

[@icosahedrondodecahedraldual]

2:32 Never in my life have I seen someone call a forest fat.

[@juliankoch9921]

When you are lost in a forest stay put. Exept when you are in central Europe. Go 500m in any direction. There will be a path or even village. If you went for an hour without encountering a village you are not in central Europe anymore. 😂 Seriously. As a child I wondered how anybody could get lost in the woods until I learned that not every place is like that

[@secondbeamship]

In the process of generations of mathematicians working on these problems many useful discoveries are made.

[@dewaard3301]

The Kobon triangle problem is very interesting because it seems computationally approachable since the exactly angles of the lines don't matter as long as the topology of the graph (the order in which other lines intersect a line) they create doesn't change. I 'conjecture' that I am wrong in this assertion.

[@nathangamble125]

To solve Lesbegue's problem, I think it's easier to think of it as "create the shape with the largest possible area, in which any 2 points are separated by no more than 1 unit length", as this will cover any shape with a diameter of 1 unit, rather than thinking about covering shapes with other shapes. The solution is either a reuleaux triangle or something very similar to it, but its exact parameters are very difficult to work out.

[@isavenewspapers8890]

Do you have a proof that that would work, or is this just your intuition? My intuition tells me that this wouldn't actually result in the right answer. The Reuleaux triangle is the curve of constant width that encloses the largest possible area, so I'm guessing that you're correct that it's the shape you're looking for. However, a Reuleaux triangle of diameter 1 is not a valid cover. This is visually shown by an illustration on this Wikipedia article: en.wikipedia.org/wiki/Lebesgue%27s_universal_covering_problem As you can see, a Reuleaux triangle is included as one of the covered shapes, without being a cover itself (and indeed being significantly smaller).

[@JohnPork-gr3bx]

9:58 visual error, should be 0.2618

[@orisphera]

LiaF reminded me of a similar problem. I saw a problem about a spaceship. I didn't solve it, but I thought of a similar problem in 2D. In LiaF terms, the forest is a half of a plane, but you know you're at the distance a from the boundary. I then changes to a different problem, found a solution for it and translated it to the 2D version of the original problem. Then, someone else came up with another solution. I successfully translated it to the different problem. It could prove that my solution was more optimal, but the calculations said otherwise

[@Illogical.]

I like convex solids a lot. More than that sentence accurately communicates on its own. But in a way that I think is, at most, vaguely relevant in the context of ulam's conjecture.

[@emory5533]

What's funny to me about these videos is that I learned a lot of this from the video game 4D Golf from the little trivia crystals. Definitely recommend if you like mini golf and geometry

[@circuitcraft2399]

The Kepler conjecture was proven true back in 1998 by Dr. Thomas Hales; it's a theorem now.

[@Velereonics]

The reason you should not go anywhere on a path is because it's likely that other people also will use the same path at some point, and they will run into you, especially if it's on a federally protected land, where there's more likely to be a park ranger who will do that trail every once in a while. This is incidentally why you should not leave trails. Unless you're extremely experienced or with someone who is or know how to navigate in the wilderness, regardless of if you know it well in this particular case or not. but as an aside gps devices are so much cheaper than they used to be, like to a ridiculous degree. you can have a little thingy that will tell you literally exactly where you are, and it probably has a function to call for help, and it sends out like a red alert to the nearest emergency service people who are trained to get people who are using gps because they have to. Even ten years ago, something like that cost five hundred dollars, unless you've had a lot of diy skills. now you can get one easily for under 100.

[@cadesmandela1935]

I would love if some random ass kid from like Timbuktu sees this video and solves it

[@ulogy]

Me: Sees video is on square packing problem. "I AM READY FOR 17 (a true classic)."

[@InflatableRockStar]

“We must define convex solids.” Me, looking at the pictures: A convex solid is a D&D dice

[@52flyingbicycles]

What’s really interesting (or frustrating, depending on your point of view) about math is that there are technically true statements in math which are nonetheless unprovable. So if something doesn’t have a proof it could just be because no one has found one yet, it’s true but no proof exists so any attempt to find a proof would be fruitless, OR you can’t prove it true because it’s false and you haven’t thought of the false proof yet. So there will always be more math to do

[@camerongray7767]

Man I loved this video, but I’ll be honest the brutal inaccuracy and inconsistency of the lines in the square packing problem was very unsatisfying. The lines didn’t touch in places the should have, uneven gaps, different thicknesses etc

[@thatdude9091]

Thank god, i was wondering how I would find my way out the forest.

[@corentinlem5333]

A little mistake when you talk about Ulam Conjecture and Kepler Conjecture : Kepler's Conjecture HAS BEEN proved ! The demo dates back to 2014 and was accepted in 2017. Stille a really good video thanks

[@acuerden]

Slightly upset your packing of balls was one of the more inefficient packings. Move the top row over 1/2 a ball width and you can push them closer together

[@Fassle]

for packing balls in space it doesn't matter whether it's a square or a triangle pattern. Both result in the same packing density, just different perspective.

[@jotch_7627]

​@@Fassleexcept in both arrangements, layers need to be offset from each other. the goal was to visualize the concept of 3D packing with 2D circles, so they should have chosen the arrangement that shows up in both 2D and 3D.

[@jazzabighits4473]

Think of it this way, things are more efficiently packed (in 3D) when they have a larger percentage of their faces touching. Spherical objects only touch eachother slightly on the faces, because they have so much curvature.

[@sideways5153]

I feel like there’s an interesting discussion to be had about whether Ulam’s packing conjecture can be disproven using a non-convex solid which is constructed out of convex solids. (Picture the silhouette of a venn diagram or an hourglass)

[@innerlichtotig9325]

it took me not joking 5 minutes to find a better packing area for 7 boxes.

[@isavenewspapers8890]

The video doesn't even show the case for 7 squares. Do you mean better than the currently known best case? Erich Friedman proved that this is impossible in 1999, so that cannot be the case.

[@gim3209]

each point on the surface of a sphere is equidistant from the center of the sphere the ratio of a sphere with diameter x to a cube with side length x is π:6 in order for ulam's packing conjecture to be disproven, there would have to be some convex 3d shape A with a ratio of its own volume (or the volume of whatever pattern of multiple As is closest to 1:1 with a rectangular prism) to the volume of a rectangular prism equivalent in length, width, and depth less than π:6, which seems obviously impossible considering the equidistance of points on the outside of a sphere

[@fancypants6062]

Great video. I think you could have left some of the example pictures on the screen for a few more seconds to give us time to see them. I had to rewind and pause a few times.

[@Bolpat]

6:33 You spelled Lebesgue wrong. There's no lesbian in him.

[@t3rg3r]

and the pronunciation was even worse

[@kingofawesomeness5375]

I think one of Ulam's friends wanted his help packing for a trip or for moving and Ulam tried to make that friend regret asking

[@kotzpenner]

Ulam seems like he’s packin‘

[@sandro-nd6ir]

Man these red squares seem so familiar

[@dewaard3301]

Yes on replacing pi with tau!

[@mecadragoon]

the first time i heard about the square packing problem i got a stroke

[@dr_cheez811]

I would not use the word approximate for the current best estimate for the square packing problem. I would say "less than or equal to" as in this is an upper bound on the solution, but it's unproven that a better packing does not exist.

[@thesuomi8550]

7:26 the second figure is a worse upper bound, not an improvement

[@kylewood4001]

JUSTICE FOR THE OPAQUE SET PROBLEM

[@hepiik.8822]

It's Stanisław Ulam, not Stanislov

[@IsmailLev]

It is translated, so incorrect in polish but makes more sense for English speakers.

[@DeuxisWasTaken]

@@IsmailLev hold on, translating your name so it's incorrect in English but makes more sense for Polish speakers. It's baffling that English still does it. We don't write "Stephen" "Stiwen".

[@Anonymous-zr8ie]

​@@DeuxisWasTakenThis is common in most natural languages. In Polish, transliteration of people's names isn't that common anymore, but we still do it - consider "Królowa Elżbieta II" (Queen Elisabeth II from Britain), or even "Chrystus" (Christ) for a biblical example. For everyday people, we also do this all the time for non-latin alphabet names, such as names in Cyrilic. Not to mention exonyms for country names - Łotwa, Holandia, Węgry, etc. Just because it's common doesn't mean it's a good thing of course, but I digress.

[@MatthewJohnson-hi2th]

Isn't like... all of this video just taken straight from the corresponding Wikipedia articles? Why don't you cite any sources?

[@MatthewJohnson-hi2th]

I've realised this is just all of your videos. To 99.9% of viewers, it is unclear where this information has come from. At no point in the video do you mention the articles you took this from. You don't even flash them up on screen. This is incredibly dishonest and disrespectful to the people who put hours into writing and editing those articles.

[@dr.palsonp.h.d815]

@@MatthewJohnson-hi2th thank you for pointing this out.

[@pokepowerz4]

the forest problem makes 0 fucking sense to me how is every answer not just a straight line

[@isavenewspapers8890]

Easy counterexample: a rectangular forest 1 unit wide and 100 units long. You don't know which way you're facing, so for all you know, you could be setting off in a straight line across nearly the entire 100-unit length of the forest without reaching the end. If you simply spiral out instead, you will reach an edge much faster.

[@coaster1235]

damn only six unsolved geometry problems left, geometers better start looking at other fields soon

[@screechingtoad2683]

Drawing a line through a donut won't always have it pass through empty space. The line would be smaller though

[@ivoirelesness]

this guy sounded like huggbees at the start ..am i wrong?

[@gljames24]

I feel like these are very similar to NP hard problems. The solution requires searching thru all possible reasonable solutions.

[@ehow5678]

If no one knows you're there find running water and follow it down stream. If you hit a lake or the ocean pick a direction

[@linoarenz548]

I love to hear about unsolved problems :)) Very interesting... It's like a treasure hunt :))

[@BigusGhostus]

i failed nearly every math class i took in high school but theres something about geometry that always interested me. only math class i got a D in first try.

[@demonmonsterdave]

Proof that our form of maths is not complete.

[@unduloid]

11:24 Tringle? Is that a kind of pringle?

[@MateusSFigueiredo]

Ulam's packing conjecture: what about half a sphere? What about half a sphere glued to a slice of a larger sphere?

[@isavenewspapers8890]

(Note on terminology: I will be calling the solid figure a ball, not a sphere. If you have an orange, the peel is a sphere, and the flesh is a ball.) A half-ball cannot possibly be a less efficient space-filler than a ball. If you just place pairs of half-balls together to form whole balls, then your packing will be at least as efficient as the ball packing. You cannot glue a half-ball to a larger half-ball to form a convex solid, so any such shape is instantly disqualified.

[@mutated_almond8335]

Love your vids

[@stephanoperra6863]

Soooo, even scientists don’t know shapes? 😭😭

[@Gwen_Hemoxia]

Humans do this for play and I think that's really cool

[@Velereonics]

I see the tau movement is still going strong

[@nathanwoodcox1788]

Bellman’s problem reminds me of that episode of the Twilight Zone, when the astronauts crash land on an “asteroid” (Nevada)

[@prod_EYES]

I think the square packing problem has to do with primes, so solving the problem would solve primes

[@mqnc6275]

Lebesgue

[@him050]

I still can’t get my head around how the Poincaré conjecture isn’t trivially simple. Like obviously any 3D shape can be made into a sphere. That’s how you know a math problem is wasay beyond the mental capacity of the average person.

[@isavenewspapers8890]

1. When you say "sphere," do you mean the 3-sphere? That's what the Poincaré conjecture is about. The object we would normally call a sphere, a round surface with a hollow inside, is a 2-sphere. This object is topologically 2-dimensional, since there are 2 degrees of freedom to move along it. It is the surface of a 3-ball, which is 3-dimensional, since there are 3 degrees of freedom within it. Similarly, the 3-sphere is the surface of a 4-ball, which is the 4-dimensional hyperball. That's what the Poincaré conjecture is talking about. 2. Not every 3D shape can be transformed into a 3-sphere. The transformation has to follow certain rules. Plus, basically every regular 3D shape that you're probably familiar with is disqualified from being included in the conjecture anyway.

[@Azumarill]

John Wetzel’s Pretzel Conjecture

[@ParkerShimoda]

I can think of at least 1 more unsolved problem. that one which asks: what shape can surround itself completely but only a finite nuber of times. heersh numbers. number 6 was solved quite recently.

[@isavenewspapers8890]

*Heesch

[@DavidAbraham-q3q]

Im trying to figure out how many brownies to put into this pan - how should i square fudge pack?

[@denieduser8655]

I don’t understand how the convex solid packing problem isn’t proven Any slight angle in the shape means less area taken by the lower solid, which can fit the solid on top of it’s slight angle Meaning the shape with the largest angles is the worst at packing, and circles have 180 degree angles and thus no even slightly obtuse angle You can’t go above 180 degrees because it isn’t convex anymore

[@ethos8863]

I"m confused by the cover thing. Isn't the smallest possible cover just the largest possible diameter 1 shape, which would be the circle, as any growth to the circle would make it larger, and any cover that could not cover the circle would not be a cover?

[@isavenewspapers8890]

No, the cover has to have every possible shape of diameter 1 fit inside, not just take up at least as much space as every shape of diameter 1. An average broom takes up less volume than there is inside of an average backpack, but you can't fit a broom in a backpack. A disk of diameter 1 is not a valid cover; for example, an equilateral triangle with side lengths of 1 has a diameter of 1, and it cannot fit inside.